Minimal actions of homeomorphism groups

• Autorzy: Yonatan Gutman
• Języki publikacji: EN

Abstrakty

EN

Let X be a closed manifold of dimension 2 or higher or the Hilbert cube. Following Uspenskij one can consider the action of Homeo(X) equipped with the compact-open topology on $Φ ⊂ 2^{2^{X}}$, the space of maximal chains in $2^{X}$, equipped with the Vietoris topology. We show that if one restricts the action to M ⊂ Φ, the space of maximal chains of continua, then the action is minimal but not transitive. Thus one shows that the action of Homeo(X) on $U_{Homeo(X)}$, the universal minimal space of Homeo(X), is not transitive (improving a result of Uspenskij). Additionally for X as above with dim(X) ≥ 3 we characterize all the minimal subspaces of V(M), the space of closed subsets of M, and show that M is the only minimal subspace of Φ. For dim(X) ≥ 3, we also show that (M,Homeo(X)) is strongly proximal.

Kategorie tematyczne

• 22F50: Groups as automorphisms of other structures
• 54H15: Transformation groups and semigroups
• 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2008
• Tom: 198
• Numer: 3
• Strony: 191-215

Daty

wydano

2008

Twórcy

autor

Yonatan Gutman

• Institute of Mathematics, The Hebrew University, Jerusalem, Israel

Identyfikatory

DOI

10.4064/fm198-3-1